Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

The nth term (a_n) of a geometric sequence can be calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

The sum (S_n) of the first n terms of a geometric series can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), where a_1 is the first term and r is the common ratio.

The sum (S) of an infinite geometric series exists if the absolute value of the common ratio |r| < 1, and is given by the formula: S = a_1 / (1 - r), where a_1 is the first term.

The common ratio (r) is the factor by which we multiply each term to get the next term in a geometric sequence.

The sum is 1.

The 5th term is 48.

If the common ratio is greater than or equal to 1, the sum of the infinite geometric series does not exist.

The 11th term is -0.0625.

The sum is 5.